Single-lane car-following is a fundamental task in autonomous driving. A desirable car-following controller should keep a reasonable range of distances to the preceding vehicle and do so as smoothly as possible. To achieve this, numerous control methods have been proposed: some only rely on local sensing; others also make use of non-local downstream observations. While local methods are capable of attenuating high-frequency velocity oscillation and are economical to compute, non-local methods can dampen a wider spectrum of oscillatory traffic but incur a larger cost in computing. In this article, we design a novel non-local tri-layer MPC controller that is capable of smoothing a wide range of oscillatory traffic and is amenable to real-time applications. At the core of the controller design are 1) an accessible prediction method based on ETA estimation and 2) a robust, light-weight optimization procedure, designed specifically for handling various headway constraints. Numerical simulations suggest that the proposed controller can simultaneously maintain a variable headway while driving with modest acceleration and is robust to imperfect traffic predictions.